Can't paste individual notes or sections of measures.

It works, with some ways.
1) Eg: you want copy-paste a quarter note: select the nothehead -> copy-> paste on a quarter rest: result as expected.
If you paste on a half rest, the note value will be identical (half note), and so on, on the whole rest (whole note), or other.

2) Other way, more recommanded/usual.
Same example, with a quarter note. Do: Shift + click on the notehead (you see a blue frame around the note) -> Copy -> Paste on a measure (possible now) , or a rest, whatever which one: the result will be always the same quarter note.

Finally, if you want only do that for a voice (let's says voice 1), open Selection Filter (F6) and untick eg voice 2 (or other) depending the content of your score. Then, same process as described above.

For voices you can use the selection filter to limit which voices you are copying. See https://musescore.org/en/handbook/copy-and-paste#selection-filter.

For parts of measures see https://musescore.org/en/handbook/selection-modes. One thing to keep in mind is that what you copy, for the most part, must be continuous. You cannot use ctrl+click to select individual notes to copy, even if they are consecutive. You have to select a section of notes like you get when you use shift+click.

If the above doesn't help, please attach the score you are having trouble with so we can understand and assist better. In general, copy / paste does work, whether for a full measure or multiple measures or a fraction of a measure, as long as it's something with a blue box around it. And as mentioned, the Selection Filter allows restricting the operation to just certain voices.